Question

已知数列{a n }满足 a 1 =1， a n = a 1 + 1 2 a 2 + 1 3 a 3 +…

已知数列{a n }满足 a 1 =1， a n = a 1 + 1 2 a 2 + 1 3 a 3 +…+ 1 n-1 a n-1 (n≥2，n∈ N * ) ，若a n =2011，则n=______．

Answer 1

由题意得，∵ a n = a 1 + 1 2 a 2 + 1 3 a 3 +…+ 1 n-1 a n-1 (n≥2，n∈ N * ) ∴ a n+1 = a 1 + 1 2 a 2 + 1 3 a 3 +…+ 1 n-1 a n-1 + 1 n a n (n≥2，n∈  N * ) ∴ a n+1 = a n + 1 n a n ∴ a n+1 a n = n+1 n ∴a n =2× 3 2 …× n n-1 =n   (n≥2) ∵a n =2011，∴n=2011 故答案为2011．